Methods and apparatus for polar to rectangular transformation



Oct. 4, 1966 H. M. MARTINEZ 3,277,290

METHODS AND APPARATUS FOR POLAR T0 RECTANGULAR TRANSFORMATION Original Filed May 11, 1959 10 Sheets-Sheet 1 FIG a 2X] i F/GLZ 4 6 8 Genelufor of Am lHude E( Periodic fime ANT) cozporcfior o ai E represeniufion of Generofor X: l(y) Tgperiod x 5x (21.). flue?) Q E =kf( min o mam ave INVENTOR. HUGO M MRT/NEZ I; BY

A7 TORNEV Oct. 4, 1966 H. M. MARTINEZ METHODS AND APPARATUS FOR POLAR T0 RECTANGULAR TRANSFORMATION 1o Sheets-Sheet z 7 Original Filed May 11, 1959 2? 55% m m m twmuo 3? 5 5: NF 23m H wwkoz m? w w m NFL wz w mm m T m INV EN TOR. HUGO M. MART lA/EZ mt. 5.0.550 2 Eu LO EOQEO a3 m x an ATTORNEY Oct. 4, 1966 H. M. MARTINEZ 3,277,290

METHODS AND APPARATUS FOR POLAR T0 RECTANGULAR TRANSFORMATION Original Filed May 11, 1959 10 Sheets-Sheet 5 FIG- 8 oui Siofic funcrion se't-up o obfuin modificolfinn of Fig-8.

LIMITER swiich 97 in posifion for-cos e,(-Tf='e 521T INVENTOR. HUGO M MARTYNE Z AT TORNEV Oct. 4, 1966- H. M. MARTINEZ 3,277,290

'METHQDS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION Original Filed May 11, 1959 1o Sheets-Shet 4 fILiMZ F/6 juncfion84 V v 7/ componen dmqe +32 2(ecomponen NOTES 1 See Fi g for definifion of 2H.)

- 2. Swiich [24 in 9; posifion for c 5 e :ITE 9 211' INVENTOR. HUGO M MART/N52 gig 2 BY "C 66 7% A T TOPNEV Oct. 4, 1966 H. M. MARTINEZ 3,277,290

METHODS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION Original Filed May 11. 1959 l0 Sheets-Sheet.5

A T TORNE Y Oct. 4, 1966 I M. MARTINEZ- 3,277,290

METHODS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION A r TOPNEV HUGO M. MARTINEZ Oct. 4,1966 H. M. MARTINEZ v 3,277,290

METHODS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION Ori ginal Filed May 11. 1959 1o Sheets-Sheet v R, Fe FIG. [8 368 1 We) R 402 406 FIG 2/ INVENTOR. (FROM FIGZOLV: W HUGO M MART/NEZ R -2v (-wv'v-- BY 3 RI A TTORNEV Oct. 4, 1966- M. MARTINEZ 3,277,290

METHODS ANDABPARATUS FOR POLAR To RECTANGULAR TRANSFORMATION Original Filed May 11,1959 l0 Sheets-Sheet 8 "FIG; 22'

INVENTOR. HUGO M. MARTINEZ A T TOR/VEV Oct. 4, 1966 H. M. MARTINEZ 3,277,290

METHODS AND APPARATUS FOR POLAR T0 RECTANGULAR TRANSFORMATION Original Filed May 11, 1959 10 Sheets-Sheet 9 FIG. 26

INVENTOR. HUGO M. MARTINEZ ATTOQVEV Oct-4, 1966 H. M. MARTINEZ 3,277,290

METHODS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION Original FiledMay 11, 1959 1Q Sheets-Sheet 10 HUGO M. MARTINEZ BY a x A T TOPNEV 3,277,290 METHODS AND APPARATUS FOR POLAR TO RECTANGULAR TRANSFORMATION Hugo M. Martinez, Chicago, Ill., assignor to Yuba Consolidated Industries, Inc., San Francisco, Calif., a corporation of Delaware Original application May 11, 1959, Ser. No. 812,566. Divided and this application Dec. 10, 1962, Ser. No.

4 Claims. (Cl. 235-497) The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

This application is a division of my copending application Serial No. 812,566, filed May 11, 1959.

This invention relates to methods and apparatus for producing physical quantities representative of mathematical functions, and more particularly representative of polar to rectangular transformation.

Computing devices and other equipment often require the generation of physical quantities such as voltages, currents, displacements, or the like, representative of various mathematical functions. Arrangements for accomplishing these purposes are commonly known as function generators. Prior art function generators often have been complicated and difiicult to construct. An object of the present invention is to provide relatively simple, reliable, and accurate apparatus and methods for generating functions, and more particularly functions representing polar to rectangular transformation.

Other objects and many of the attendant advantages of this invention will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings wherein:

FIG. 1 is a graph illustrating a periodic time representation of a linear function having a duty cycle less than 100%;

FIG. 2 is a block diagram showing one apparatus of this invention;

FIG. 3 is a graph illustrating a periodic time representation of a linear function having a 100% duty cycle;

FIG. 4 is a graph illustrating a periodic time representation of the monotonic segment of the sine function having a 100% duty cycle;

FIG. 5 is a block diagram of an apparatus for generating the arc sine function;

FIG. 6 is a schematic diagram of an apparatus for generating the sine and cosine functions using the apparatus of FIG. 5 in the feedback of an amplifier;

FIG. 7 is a block diagram of an apparatus for generating the sine and cosine functions using the basic method of the invention;

FIG. 8 is a graph illustrating a periodic time representation of the arc sine function produced from a sine wave;

FIG. 9 is a graph illustrating the static function set-up used to modify a sine Wave to produce the graph of FIG. 8;

FIG. 10 is a schematic diagram of an apparatus, using an arc sine generator in the feedback of an amplifier, for producing the sine function and cosine function for a range of angles extending over 31r radians;

FIG. 11 is a graph illustrating the operation of the means in FIG. 10 for extending the usable angular range;

FIG. 12 is a schematic diagram of an apparatus using the basic method of the invention for generating the sine I and cosine functions with means extending the angular range over 31r radians;

FIG. 13 is a schematic diagram of an apparatus used United States Patent generating a periodic time representation of a positive exponential;

FIG. 18 is a schematic diagram of an apparatus using the apparatus of FIG. 17 in the basic method of the invention for generating a logarithmic function;

FIG. 19 is a schematic diagram of an apparatus using the apparatus of FIG. 18 in the feedback of an amplifier for generating an exponential function;

FIG. 20 is a schematic diagram of an apparatus for generating essentially a periodic time representation of a negative exponential;

FIG. 21 is a schematic diagram of an apparatus using the apparatus'of FIG. 20 and the basic method of the invention to generate the logarithm of reciprocals;

FIG. 22 is a schematic diagram of a circuit using the apparatus of FIG. 21 in the feedback of an amplifier for producing negative exponentials;

FIG. 23 is a schematic diagram of a circuit using the apparatus of FIG. 18 for producing positive constant powers of a variable;

FIG. 24 is a schematic diagram of a circuit using the apparatus of FIG. 18 and of FIG. 21 for generating negative constant powers of a variable;

FIG. 25 is a schematic diagram of a circuit using the apparatus of FIG. 18 for generating variable powers of a variable;

FIG. 26 is a schematic diagram of an apparatus for producing periodic time representations of a linear function, a quadratic function, a cubic function, etc.;

FIG. 27 (a) is a schematic diagram of a circuit using the apparatus ofFIG. 2-6 for generating square roots;

FIG. 27(b) is a schematic diagram of a circuit using the apparatus of FIG. 26 for generating cube roots;

FIG. 28 (a) is a schematic diagram of a circuit using the apparatus of FIG. 27=(a) to generate squares;

FIG. 28(b) is a schematic diagram of a circuit using the apparatus of FIG. 27(b) for generating cubes; and FIG. 29 is a schematic diagram of an apparatus using trigonometric relations to generate constant powers without the use of logarithms.

value, or values, when special values are assigned tocertainxquantities, called the arguments or independent variables of the function. one variable, x, are the following: 2x; (1-x sin x; e; log x. These are also called functional expressions. -O ne quantity is said to be a function of another if to each value of the second (the independent variable) there corresponds a value of the first (the dependent variable). The range of the independent variable is either explicitly stated, or understood from the context. The foregoing examples of functional expressions are specific functions of x. The symbols used for a general function of x are f(x), g(x), F(x), (x), etc. Such symbolsare used when making statements that are true Examples of functions of for several different functions, in other words, statements that are not concerned with a specific form of function. Frequently a single symbol, constituting the independent variable, is used to represent a function and is then defined as equal to the particular, specific functional expression in the dependent variable or to the general function. Thus, for example, where the symbol y is used to represent a function it may, using the previous expressions as examples, be defined specifically as y=2x; y=(1x y=sin x; etc., or it may be defined in the case of a general function as y=f(x); y=g(x); etc.

An inverse function or the inverse of a function is the function obtained by expressing the independent variable explicitly in terms of the dependent variable and considering the dependent variable as an independent variable. If y=f(x) results in x=g(y), the latter is the inverse of the former (and vice versa). Thus where a function y is defined as y=2x, the inverse function is x= /2y. In the case of the general function where y=fi(x), the inverse function is written x=f- (y).

It must be remembered that a function is always regarded as being confined within limits constituting the range of interest. That is, there are limiting values to the function which depend on either explicitly expressed limits of the dependent variable or, impliedly, those limits of the dependent variable for which the function is defined.

A function generator is an apparatus which, assuming the functional relation between two variables, for example, to be expressed by y=fi(x), will, when supplied with any particular value of x, say x within the limits of the function produce the corresponding value of y, say y This process of producing from a given value of x the corresponding value of y is called generating a function.

Denoting in general a functional relation between two variables by y=f(x) and the inverse function by x=f (y), the method of the present invention achieves the automatic physical realization of the relation y=f(x) by the use of the relation x=f (y). This means that given a specific value of x in some physical form such as a voltage, current, or the like, then the corresponding value of y will be generate-d in the same or analogous physical form, using the relation x=f '(y). It is noted again that while in the relation y=f(x), x is the independent variable .and y is the dependent variable, the reverse is true in the inverse relation x=f -(y). As a specific example, if y=arc sin x corresponds to y= (x) wherein f(x)=arc sin x, then x=sin y corresponds to x=f (y) and (y)=sin y.

Prior art automatic generation of functions by the use of given inverse functions has been accomplished by automatically solving the equation xf- (y)=o using y as the unknown. This system is explained on page 340 of the book Electronic Analog Computers by G. A. K-orn and T. M. Korn, published by McGraw-Hill Book Company, .New York, second edition, 1956. The practical success of such equation solving methods is largely dependent on the ease with which f- (y) can be generated. By generation of ;f (y) is meant that given a value of y, the corresponding value of f- (y) is produced. These methods all give static representation of f (y), wherein y is time independent.

In contrast to the foregoing automatic equation solving method, the method of the present invention does not rely on the solving of equations; and instead of a static representation of f- (y) it uses a dynamic representation or time representation of (y) by, in effect, replacing y with real time, in which replacement an interval of time represents the range of y. To understand this method, an explanation of certain terms in appropriate. A time representation of a function =g(x) defiined for x x x can be accomplished by letting an interval of time correspond to the range of x from x to x and generating the function =g(x) as a function of time over this interval. Specifically, a trans- 4 formation of x to the time domain is made by the linear relation x=kt+x wherein t=0 is the instant of time defining the start of the time interval referred to above. The size of the time interval is given by x x /k. It must be noted that in each specific instance where the linear transformation itO the time domain is accomplished the variable t is limited in its range from zero to x x /k.

For example if =2x, then a graph of the function within its necessarily prescribed limits of x and x would be a straight line in the b, x coordinate system wherein is the ordinate and x the abscissa and the end points of the straight line would have ordinate values of 2x and 2x When the linear transformation to the time domain is accomplished, the abscissa becomes t and the equation becomes =2(kt+x A graph of this latter equation in the t coordinate system yields a straight line whose endpoints again have ordinate values of 2x and 2x The distance between the projections on the t axis of the endpoints is x x /k since the abscissa of the lower limit of the function is t=0 and the abscissa of the upper limit of the function is t=x x /k. The graph thus terminates very certainly at points determined by the region of interest, although t, the independent variable, representing real time, of course continues indefinitely and therefore =2(kt+x could ostensibly be plotted as a line indefinitely long.

For purposes of this invention a regularly repeated time representation of the function is required. This is called a periodic time representation of the function. In general it is not practical to write an equation for a periodic time representation, although in specific cases it may be simple to do so. The equation above, =2(kt-|-x represents the actual equation of only one portion of one cycle of the periodic representation, namely, that portion of one cycle which exhibits the functional relationship between a dependent variable and an independent variable exemplified by the equation =2x wherein x is considered to lie only between x and x and wherein, correspondingly, varies only from 2x to 2x FIG. 1 shows one example of a periodic time representation of the function =2x. This graph would be said to represent the functional relation =2(kt+x in the region from i=0 to t=x x /k but it must be observed that in fact this functional relation holds only for the segments from a to b, from d to e, from g to it, etc., and then only if t be regarded as starting at zero at each low terminus, i.e., at a, again at d, again at g, etc. The segments of the graph from b to c, from c to d, from e to from f to g, etc., are not repersented by the equation =2(kt+x From the foregoing it is clear that a periodic representation of a function involves displaying the function repetitively in time in such a manner that equal intervals of time correspond to the range of the independent variable. Thus in FIG. 1, which illustrates a periodic time representation of the functional relation =2x wherein x, the independent variable, ranges from x to x the interval of time represented by the lengths ac, df, etc. corresponds to the range of x from x to x From the graph in FIG 1 it is seen that the basic period of the graph, T, is represented by the lengths ad, dg, etc. As shown, only a portion of each basic period of the time representation is occupied by the function, e.g., the time intervals represented by abscissa lengths ac, df, etc. The functional relation is not being represented during a portion of each period shown as the time intervals bcd, efg, etc., each of which has a duration T-x x k.

The present invention can use periodic time representations of the type shown in FIG. 1 wherein the repeated representations of the functional relation of interest are separated by a line on the graph representing a value or values not essentially of interest. The invention can also use another type of periodic time repersentation wherein the functional relation of interest effectively occupies the entire period of time under consideration, This other type of periodic time representation falls in two categories: 1) Where the repeated representations of the functional relation of interest are contiguous, and (2) where the functional relation of interest is contiguous to and alternates with its mirror image. This latter type of periodic time representation is the most common and the simplest to use and to understand in its behavior in the practice of the invention. The former type, exemplified in FIG. 1, is sometimes more convenient to produce. An explanation of the generation and use of this former type in the invention is set forth hereinafter in relation to the embodiments of FIGS. 26 and 27.

The common term for a device which gives a periodic time representation of a function is a wave form generator. In contrast to this, the term function generator implies a device such that if a value of an independent variable is introduced, the device produces the corresponding functional value. The independent variable may or may not be varying with time. If the wave form generator produces a periodic time representation wherein each functional display follows its predecessor immediately with no dead interval between them, the periodic time representation is said to have a 100% duty cycle. In FIG. 1 if the abscissa intervals cd, jg, etc., were each reduced to zero the representation would have a 100% duty cycle. The actual duty cycle of FIG. 1 is given by x -x /kT percent.

The useful terms for describing this invention having been defined, a proof and explanation will now be offered of the novel mathematical relation on which are based the method and apparatus of this invention.

A RELATION BETWEEN A FUNCTION AND ITS INVERSE Given a function (x) defined for a x b, let y denote a particular but arbitrary value of the dependent variable y. Define the variable E as A1 213% 2 if y yo 6 Then,

o f (yo) The following is concerned with establishing general formulas for E for a large class of functions of practical importance. Interest thus centers upon the integral b L ty, yo) (1) Evaluation of this integral depends, of course, upon the nature of f(x), but attention is here restricted to cases where the integration is, in the first place, possible. Consider next, then, the values of x in the interval [0, b] which corresponds to y=y that is, the set of x such that f(x) =y Let x l and x denote, respectively, the minimum and maximum values of x in the set j- (y Then XoM xon Xom Imposing the further restriction that for no x is (x a maximum or minimum, then, with n even, the first and last integrals of (2) will have the same integrand values, that is, both A or both A If n is odd, one will be A and the other A We may, therefore, write for n even,

and for 11 odd,

ySyo a.

y yu b f Edx: a

Collecting terms and simplifying, the general formula for can be Written as l y 31/0 T A2 if y yo and e(x -a). In the event that y is a maximum or minimum for one or more x must be treated as two points with a corresponding increase in the value of n. Formulas 4a through 4d will then apply.

It may be noted that results in the several foregoing analyses would be substantially the same if E were alternatively defines as where K is a constant of any finite value.

Examples (1) y=x a x b.Here f (y =y and since there is only one real valued root for y n=1. Further, ;f(x )=f(x e) 0 for every x because x is an increasing function. Formula 4c is therefore applicable and leads to the expression b L Eo yo d =(A1A2 a Alb-Ala Dividing through by (ba) to obtain E ave= 1 2) D+ z 1 Letting A =b and A =a, then E =x f( ol) f( o1' hence, Formula 4b is applicable, resulting in ave i r- 1) o1 o2) 2' Now using the fact that x =x and choosing A =a, A 0, leads to E x,, /y

(3) y=sin x, o x 21r.Since f" -(y )=arc sin y then n=3 if y =0, and n=2 if 3 0. It is recalled here that values of x for which y is a maximum or minimum (when y =i1) are each to be treated as two points.

(a) For y O, relation 4a holds.

since x =7T o1 If we choose A =1r/2 and A =1r/2, then (b) For y 0, relation 4b is applicable. Hence,

since x =31rx,,

Again letting A =1r/2 and A =-1r/2 as in case (a),

=1r-X the supplement of x (c) For y =0, relation 4c is used. Therefore,

1 nve ;i( l 2) o1 %2+ o3) 2] Once again, letting A =7r/2, A =7r2 and noting that x =0, x =1r, 1: :21, the result is The significant result in this example is that by letting A =1r/Z and A =-1r/2 for all three cases corresponding to y O and y 0, the value of E in each case corresponds to a correct, but numerically smallest member of the set are sin y One can therefore write:

E =arc sin y -1r/2 E,, 1r/2. This result is extensively employed in the section on illustrative applications.

(4) Monotonic functi0ns.Monotonic functions with single valued inverses are readily handled by Formula 40 if the function is increasing (Example 1) and by 4d if it is decreasing. If the function f(x), a x b, is an increasing one, it follows from 40 that choice of A =b and A =a makes E =f (y On the other hand, if f(x) is decreasing, Formula 4d indicates choosing A =a and A =b to make E (y None of the Formulas 4a through 4d are applicable however, to monotonic functions with multiple valued inverses because they were developed by assuming all sets f- (y) to be finite. This condition of finiteness does not hold for monotonic functions with multiple valued inverses. Appeal must therefore be made to Equation 2. The integrand of the second integral on the right will be the same as that of the first, leading to b 0 b f Edx=f A dx +f A da: for a non-decreasing func- 3 a XOM tion and to b x0 b I Eda: ==f M A dx A dz for a non-increasing funca a oM tion Hence,

If for f(x) non-decreasing the choice A =b, A =a is made, and for f(x) non-increasing A =a, A =b, then in either case E =x =maximum member of the set f (yo)- In the foregoing examples it was shown that the appropriate choice of values for A and A in the variable E(y, y leads to an average of this variable which is equal to the least of the inverse values f- (y The invention uses this mathematical principle for the following method of generation of a function of a single variable f(x). By generation of a function of a variable is meant the production of a physical quantity such in voltage, current, electrical resistance, mechanical displacement or the like whose magnitude varies in accordance with the variation of the [function of the variable.

METHOD OF FUNCTION GENERATION Object: To generate y=f(x).

Step 1.-Generate a periodic time representation of the inverse function f" (y).

Step 2.--Compare the amplitude of this time function with a given 'value x of the independent variable of the required function f(.x).

discontinuous function A, if at so, {A if x x where the values of A and A are time independent, or at least do not vary appreciably over a single period of 1 Step 4.-Take the time average of E(x, x This time average, for appropriately chosen values of A and A is proportional to the value of the dependent variable y corresponding to x=x It should be noted that the value of x, namely x is permitted to change only at a rate which is much smaller than 1/ T, the repetition rate of the periodic time representation of the inverse function f- (y). Also, it should he noted that if the function f- (y) happens to be inherently periodic, its period need not correspond identically with the period, T, chosen ifior the periodic time representation of f (y) in the method of this invention. For example if f- (y)=sin 0, its period would normally be regarded as 211' radians, constituting the length of the shortest equal sub-interval into which the range of the independent variable, 0, can be divided and obtain exactly the same graph of the function in each sub-interval. However, in practicing the method of the invention, wherein it is required to present a periodic time representation of f* (y)=sing 0, which involves the substitution of (kt+0 [for 0, it is possible within the scope of the invention to choose a period T for vthe tunction sin (kt-F0 corresponding to a range of 0 over only 11' radians. of f- (y) would preferably be made up of a repetitive presentation in regular sequence of only that generally S-shaped portion of the ordinary sine graph lying between- GENERALIZATION OF BASIC METHOD OF FUNCTION GENERATION The symbolic expression in Step 3 of the aforementioned method implies at first blush that it is required to generate (1) E=A during the time interval, say A, throughout which x x and (2) E==A during the time interval, say 6, throughout which x x However, since Step 4 requires taking :a time average of E, it should be apparent to those skilled in the art that exactly the same end result will be obtained if (1) E is caused to have the value A not during the time interval, A, wherein x x but during a different time interval, say A, so long as A=A; that is, so long as the length of time in the interval A equals that in the interval A; and

l (2) E is caused to have the value A notduring the interval, 6, wherein x x but during a different time interval, say 5' so long as 1?:6.

Referring the explanation for simplicity to the occasion of a single time representation of the inverse functional relation, the significant fact is that the two values A and A of E divide between them an interval of time equal to the total length of time during which the time representation of the inverse lfunctional relation occurs. Actually, the generation of E need not even be simultaneous with the time representation of the inverse lfiunctiona-l relation although in practice it is. The share of time interval assigned to A is equal to the length of time that x x and the remainder of the time interval is assigned to A However, it is totally immaterial to the value of the end result, namely the time average of E, whether A takes its share from the first portion of the time interval or from the last portion of the time interval or from the middle portion of the time interval or partly from two or more such portions.

1 From the foregoing it is clear that the following is a In such a case the periodic time representation.

10 I GENERALIZED STATEMENT OF THE -METHOD OF FUNCTION GENERATION I Object: To generate y=f(x).

Step 1.-Generate a periodic time representation of the inverse function f- (y).

Step 2.-Oompare the amplitude of this time function with a given value x of the independent variable of the required function f(x).

Step 3.Generate, as a result of Step 2, a discontinuous function A, during an interval of time equal to that When a: 3x E0131 0) A, during an interval of tune equal to that When a 0 where the values of A and A are time independent, or

at least do not vary appreciably lover a single period of f (y)- Step 4 .Take the time average of -E (x, x,,). This time average, for appropriately chosen values of A and A is proportional to the value of the dependent variable y corresponding to x=x It should be noted that the more extensively Verbalized expression E in Step 3 immediately above is fully equivalent to and interchangeable with the more succinct, pre-' dominantly symbolic expression in Step 3 of the earlier recitation of the method. Although the predominantly symbolic expression, 'being more convenient to write, will be generally used hereinafter, it must be understood and interpreted always to include the generalized expression.

FIG. 2 shows diagrammatically one apparatus of the present invention for carrying out the aforedescribed method of function generation.

Numeral 2 indicates a generator of the periodic time representation of x=f- (y). The output of this generator, being, for example, a voltage or the like, represented by the expression x(t+T), is fed into an amplitude comparator 4 into which is also fed the physical quantity such as voltage, representing x the given value of x for which it is desired to produce the corresponding value of the function of x. The amplitude comparator 4 compares the value of x with the value of x generated by the generator 2 as that value of x varies within the region of interest during the time cycle. During the period of time when x, the output of generator 2, is less than or equal to x the amplitude comparator 4 puts out a first signal and during the time while the value of x fed into the amplitude comparator exceeds the value x the amplitude comparator puts out a second signal. The auxiliary function generator 6 generates the discontinuous function E, which function has two values, one value being produced by the generator 6 when the generator 6 is receiving the aforementioned first signal from the amplitude comparator 4, and the other, when the generator 6 is receiving from the amplitude comparator 4 the aforementioned second signa The output of the generator 6, which again may be an electrical quantity such as a voltage, is averaged by an averaging device indicated. by the numeral 8. When voltagesor currents are in-' volved, such an averaging device can be constituted by a filter. The output of the averaging device 8 is simply the average value of the auxiliary function E and represents, when the proper magnitudes have been chosen for the two discrete values of E, the value y of the function of x corresponding to the value x of the independent variable.

ILLUSTRATIVE APPLICATIONS (1) Generation of x= /2.--To illustrate the use of the method of this invention, let it be desired to generate spondingly x x x The inverse function is =2x. Applying the method of the invention, a periodic time representation of =2x is generated. One such periodic time representation is shown in FIG. 3 which happens to.

have effectively a 100% duty cycle. The amplitude of this time function is compared with a given value of the independent variable of the required function x= /2. Thereupon there is generated, as result of the comparison, a discontinuous function 1 if So A2 o In this example A is assigned the value x and A is assigned the value x The auxiliary variable E over one cycle has the value x during the time interval OP and has the value x during the time interval PQ. The time average of E is then taken over the cycle and this time average will be the value x of .the dependent variable x in the original function relation corresponding to In FIG. 3 the scale chosen at random happens to have the following values: x =1; x =4; OP=2; PQ=6. Thus, 14 :4; A =1; and the time average over one cycle is given by:

To check the validity of the method, is shown to be 3 /2,

The generalized concept of the basic method of the invention applied to the generation of x= /2 can be seen from the following. In FIG. 3, let there be established on the t axis a point P located between P and Q, such that 0P=PQ. Then, let the generation of the auxiliary variable take place in such as manner that E assumes the value A =x during the interval P'Q and assumes the value A =x during the interval OP. Since, under this concept, the two values A and A of the auxiliary variable E have divided betwen them the total time interval OQ of the cycle of the time representation of the inverse functional relation in the same proportion that they did in the former case, when A -=x occupied the interval OP and A =x occupied the interval PQ, then it is apparent that the average of E over the full cycle will be exactly the same as in the former case, and will equal x In this instance, E has the value A not during the interval of time, OP, when but during the interval of time PQ=OP. Similarly, E has the value A not during the interval of time PQ when q p but during the interval OP'=PQ. In actual practice with electronic equipment, it is often more convenient to use an arrangement exemplified by this latter case, wherein A is generated during the interval P'Q. This is particularly true when the time representation of the inverse functional relation is symmetrical about its intercept on the abscissa axis such as the sine time function shown in FIG. 4. In such a case, the sum of the time representation of the inverse plus the given value of the independent variable changes sign at the point corresponding to P and this change of sign is useful to control the auxiliary function generator.

It is apparent that, in principle, the method of this invention can be practiced by generating only a single cycle of the time representation =2(kt+x This will produce a precisely correct value x of the function x= /2 so long as remainsfixed during the single cycle.

If remains fixed over a plurality of cycles of the time representation, the average of E over all these cycles will still be precisely x If E is averaged over many cycles, say some thousands of cycles, it will remain indetectibly different from x even though the comparison of with the 5 of the time representation be caused to cease at some instant prior to the exact completion of the last full cycle of the time representation. Since, in practice, it is commonly required to generate values of a dependent variable corresponding to numerous values of an independent variable it is, in practice, desirable to produce a limiting values of +1 and 1.

periodic time representation of, e.g., =2(kt+x so that there will always be at hand a contemporary cycle of this time function against which to compare an existing value of b so as to generate promptly the auxiliary variable E and hence the ultimately desired value x That is, the most usual case is the one where takes on various values as time progresses and does not remain fixed at one value.

If (p changes discontinuously to a new discrete value, say the corresponding value x could be generated by merely generating one additional cycle of the time representation =2(kt+x and performing the comparison and generation of E as in the first case. However, as just previously indicated, it would usually be desirable in conventional computers to produce a periodic time representation, i.e., a continuous repetition of the cycle, inasmuch as usually will change with time and, moreover, will usually change continuously with time. So long as the value of 5 remains substantially fixed during one cycle of the time representation =2(kt+x the generated function will be substantially x Stated in other words, must for accuracy change at a much slower rate than the repetition rate of the periodic time representation. If, for example, were itself subject to a periodic variation, then, for accuracy, the frequency of the variation of qs should be much less than that of the periodic time representation =2(kt+x In practice, if 1/ T is the repetition rate of the periodic time representation, 1/100 this rate or 1/ T is usually the maximum rate at which will be allowed to change to achieve practical computing accuracy. The slower the change in (p the more accurate will be the corresponding value of x that is produced.

(2) Generation of 0=arc sin x.The inverse function is x=sin 0. The sine is an'inherently cyclic function with A convenient range for consideration of the function 0=arc sin x is for 1r/2 01r/2 since this corresponds to the range -1x1 yielding a sample extending over the entire possible range of the sine. The elementary obvious seg ment of a sine curve to be used for exhibiting a periodic time representation of the inverse function x=sin 0 would be the region where 0 ranges from 1r/2. to +1r/2 and the equation of one cycle of such a representation would be x=sin (kt+0 where 0 =1r/2 and 0 =1r/2 so that t varies from t=0 to The period of such a cycle is 1r/k01r/k. FIG. 4 shows a periodic time representation of this sine function using the elementary segment from 1r/2 to +1r/ 2 as the basic constituent. The generation of 0 =arc sin x for any given value x of x is accomplished in accordance with the teaching of the invention viz. by comparing this segmentary time representation over a cycle with x and generating the auxiliary function and then averaging E over the cycle. As mentioned in the preceding illustrative application, the comparison and averaging can just as well take place over a plurality of cycles of the time representation and will give the same accurate result. Also, if x changes with time, the only practical application of the invention is by the use of a repetition of the cycle of the time representation and this repetition must for accuracy be at a rate much faster than the rate of change of x.,.

The generation of the wave form illustrated in FIG. 4, constituting a repetition of the segment of a conventional sine wave lying between 1r/2 and +1r/2, is certainly possible and can be accomplished by methods well known in the art as explained, for example, in the volume entitled Waveforms, No. 19 of the Massachusetts Institute of Technology Radiation Laboratory Series published in 1949 by McGraw-Hill Book Co., New York. However, it is readily apparent that each full cycle of such a wave form constitutes one symmetrical half of the conventional full sine wave cycle lying between 1r/2 and 31r/ 2. It is further apparent that, because of the symmetry, the average value of E obtained by comparison of x with that half of the conventional sine wave lying between 1r/2 and 31r/2 would be identical with that obtained by comparison of x with the segment of a sine wave lying between 1r/2 and 1r/2. Therefore it is clear the same identical accurate result obtained by the use of the wave form of FIG. 4 can be achieved by using a full sine wave form. The full sine wave form is easily generated by means of a sine Wave oscillator and would normally be less expensive to ue than the wave form of FIG. 4.

The use of the entire full wave output of an ordinary sine wave oscillator to generate 6=arc sin x is now described. As previously noted, the inverse function is x=sin 0. Using conventional symbols, a periodic time representation of the inverse function employing the full wave is obtained by setting 0=wt, where t=time and w=angular frequency. The function x=sin wt is, as noted, easily generated by means of a sine wave oscillator. Next, the output of the sine wave oscillator is compared with a given value of x, denoted by x and as a result of this comparison, there is generated the auxiliary function Because of the previously mentioned symmetry of a sine Wave, the average of E over one cycle of sin wt will be the same as the average of E over that portion of the cycle lying between 0=-1r/ 2 and 0=1r/2 and furthermore the average of E over one cycle will be the same Whether the cycle starts at x=1 or x=o or elsewhere. Moreover, if the average of E is taken While x remains substantially unchanged during many cycles of sin wt, the value of E will be substantially the same even though the comparison of x with x is terminated before the exact completion of an integral number of cycles of sin wt.

Since E(sin wt, x is a periodic function of period 21r/ w, its time average over a plurality of periods is the same as that over a single period. This average has already, in effect, been obtained in Example 3 above; and as before, there are three cases to consider: x 0, x 0 and x =0. For x 0 we have (see FIG. 5):

1 Where t are sin x E(sin wt, :0

=wt =arc sin x as required The cases x 0 and x =0 are treated in a similar manner, all leading to the result that E =arc sin x Thus, by appropriate filtering of E(sin wt, x to obtain its time average, the value of arc sin x is generated. Since x was an arbitrary value of x within its range of definition, the function 0=arc sin x, 1r/26 1r/ 2 is obtained.

The physical schemes for carrying out the generation of 0=arc sin x, as in all the following examples, are very numerous depending on the nature of the variables and the speed and accuracy requirements. One such scheme where the variables are voltages, as in electronic analogue computers, is shown schematically in FIG. 5. A voltage, representing sin wt, supplied by any convenient sine wave generator, is applied to the terminal of an amplitude comparator 12. The amplitude comparator 12 can be of any convenient form known in the art. Amplitude comparison and various types of comparators are described in the aforementioned volume Waveforms, especially in Chapter 3 and Chapter 9. A voltage representing x is supplied to terminal 14 of the comparator. The output of the comparator 12 has two values: one if the comparator has found that x x and the other if x x The output of comparator 12 is fed to the generator 16 of the auxiliary function E. The output of comparator 12 causes auxiliary function generator E to select one or the other of its two input voltages representing 1r/ 2 and -1r/ 2. It selects the former if x x and the latter if x x The output E of generator 16 is then a discontinuous function having the two values constituted by the voltages representing 1r/2 and 1r/ 2. To average E this output is fed through a low pass filter, with cutoff below the frequency w, which effectively takes a time average of E so that the output at terminal 20 of the filter 18 is E which, as previously demonstrated, equals arc sin x A compact electrical arrangement of the embodiment of FIG. 5 can be made by joining together in one unit the comparator 12 and the auxiliary function generator 16 wherein a polarized or differential relay is used, operated by the combination of the voltage at 10 and the voltage at 14 to make contact alternatively with a source of 1r/2 voltage or a source of 1r/2 voltage. Mechanical comparators embodying the invention include any of the various forms of differential distances or angle detectors such as dilferential gears. Electronic comparators and switching circuits would preferably be used when the invention is used in a high speed computer.

Although for simplicity of explanation the input to terminal 10 of comparator 12 was shown as sin wl, nevertheless in practice, particularly in conventional electronic computers, it iscustomary to use voltages of say volts to represent the limiting values of the range of a variable. Thus, more generally, the input at terminal 10 would be shown as say x=A sin wt where A might be 100 and A sin wt would be the actual instantaneous voltage at 10- In such a case A x A. Similarly, the inputs at terminals 22 and 24 would more generally be designated as k1r/2 and k1r/2. The actual voltage from filter 18 would then be I E =k are sin However, multiplying factors are readily removed as inserted by conventional procedures and the actual value of the function can thus always be extracted.

(3) Generation of 0=arc cos x.4ince arc cos x=arc sin x1r/ 2, it s-uflices to add 7r/ 2 to the arc sine function in order to obtain the arc cosine function. Thus can be done in the circuit of FIG. 5 by adding 1r/2 to the output of generator 16 or to the output of filter 18. If't'he arc cosine function is desired it can readily be produced in the conventional manner kown in the art by feeding --arc cosine into an operational amplifier, the output of which will then be are cosine. The range is -ar=00.

Of course 0=arc cos x can also be generated by the use of the method of the invention directly without recourse to a modification of the arc sine generator. This could be done by an apparatus similar to that of FIG. 5 wherein the inputs to comparator 12 would be cos wt and x and the inputs to generator 16 would be Jr and 0 instead of 1r/2 and --1r/2. It should be noted that cos wt is, of course, identical in form to sin wt and therefore is obtained from an ordinary sine wave oscillator, which can, as well, be called a cosine wave oscillator. The function then generated by generator 16 would be {7r if x x This is for the range 06661.

(4) Generation of sin 0 and cos 6.--This can be done for sin 0 in one of two ways: (A) by placing the arc sine 15 circuit of FIG. 5 in the feedback of an amplifier; or (B) by a direct application of the method of the invention. Both methods are easily adapted to the generation of cos 0. Method A is illustrated in FIG. 6 and Method B is shown in FIG. 7.

In FIG. 6 numeral 25 designates an arc sine generator identical to the entire assembly of FIG. 5 which receives sin wt at one input terminal 26 and receives y at its other input terminal 27 and yields are sin y at its output terminal 28. The output of the arc sine generator, and a voltage representing 0, applied at input terminal 29, are each fed through separate identical resistors R to the sum ming junction 30 of an operational amplifier 32. The output of this amplifier at 34 will be a quantity such that its arc sine equals +0. This quantity is then sin 0. This arrangement is operative in the region from 1r/2 to 1r/2. By throwing the switch 36 from the zero position to the position where 1r/2 is fed into summing junction 30 through another resistor R, of the same value as each of the aforementioned two resistors, the output of the apparatus becomes sin (6-1r/2) which equals cos 0. If cos is desired, it is a simple matter to feed the output at 34 into an amplifier to reverse its sign. It should be noted that the range of the device of FIG. 6 when used to generate a cosine function is from O fl rr.

In FIG. 7 an apparatus using the direct application of the method of this invention is shown. A comparator 38 is supplied at terminal 40 with z(t) a periodic time representation of the arc sine function of the variety shown in FIG. 8, for example. The voltage representing 0, whose sine orcosine is ultimately to be produced, is fed into terminal 42. The comparator compares the two voltages at terminals 40 and 42 and then actuates auxiliary function generator 44, which is supplied with voltages at terminals 46 and 48 representing +1 and l, so that generator 44 generates The output of generator 44 is averaged by running it through a low pass filter 50 whose cutoff is below frequency 1/ T but high enough to have little effect on the maximum frequency of change of 0. The output of filter 50 at terminal 52 is then y=sin 6 where 1r/2 6 1r/ 2. By throwing switch 54 from the zero terminal to the 1r/2 terminal, the independent variable input to the comparator becomes 91r/2 instead of 0 and the device will be made to produce y=sin (61r/2)=cos 0 where (l fi m As previously mentioned cos 9 can easily be converted into cos 0 by feeding it through an amplifier.

The periodic time representation of the arc sine function can be obtained in a variety of ways for use in Method B. Among these are:

(a) Harmonic synthesis of time sine functions which is simply the reverse of harmonic or Fourier analysis;

(b) Harmonic modification of a square wave which amounts to filtering out from a square wave (which contains practically all frequencies) such frequencies that those which remain produce the desired time function;

(e) Letting the x input in FIG. be a triangular wave form of amplitude +1 and 1 and of repetition rate much less than w. That is, x can be varied as a triangular function of time and the output of terminal 20 of such a device as FIG. 5 would then be a periodic time representation of arc sin x (d) Direct modification of a periodic time function, such as sin wt, with a diode function generator. The last mentioned item is shown in FIG. 8 where sin wt is being modified to a time function that gives the values of the arc sine between 1r/2 and 1r/2 in a periodic manner. FIG. 9 shows the static function that would have to be set up on a diode or similar function generator to so modify sin wl. The use of diode function generators and the like in this manner to accomplish modification of functions is fully set forth in Korn and Korn op. cit. page 290 if.

As previously noted, in the illustrative sine and cosine generators of FIGS. 6 and 7, the range of 0 is 1r/2 to 1r/ 2 for sin 0 and O to 11- for cosine 0. These ranges can be extended by appropriate modification of the equipment when 6 exceeds these ranges. One example of an actual circuit exhibiting such a modification is shown in FIG. 10. This circuit can be said to represent essentially an actual circuit exemplifying the schematic arrangement of FIG. 6 plus the modification employed to extend the range of 0 to from 31r/2 to 31r/2 for sin 0 and to -1r to 211' for cosine 0. The circuit comprises a comparator including an operational amplifier 54 having two input terminals 56 and 58 into which are fed, respectively, sin wt, and y for comparison. The limiter connected to amplifier 54 is arranged to produce at the output terminal 60 a discontinuous voltage function having only two values, say +2 if sin wt+y 0, and 2 if sin wt+y 0. This voltage is chosen as being sufficient to cause diode 62 either to conduct or not to conduct. The circuit further comprises an auxiliary function generator and an averaging device for its output including diodes 62 and 64, operational amplifier 66 with input terminals 68 and 70, filter circuit 72 and output terminal 74.

If sin wt y, the plate of diode 62 is made negative and therefore diode 64 will conduct and the net voltage appearing at terminal 76 will be that due to 71'/ 2 from terminal 68 minus, from terminal 70, 1r/2 increased by virtue of r /2 to 1r so that the net effect at terminal 76 will be that of rr/Z. When sin wt y, the net voltage at terminal 76 will be that due to effectively +1r/ 2. The output at 76 is averaged by the filter circuit 72 so that are sin y appears at terminal 74.

To produce the sine of 0, it sufiices to embody the aforedescribed are sine generator in the feedback of an amplifier circuit in the manner of FIG. 6. In FIG. 10 the output 74 is placed in the feedback of operational amplifier 78, whose output at terminal provides the y to be fed into the arc sine generator at terminal 58. 0, whose sine it is desired to generate, has its negative applied at terminal 82 and joins the output of the arc sine generator at summing junction 84 serving as the input source for amplifier 78. Since the entire monotonic section of the sine curve is reppresented by the portion lying between 0=1r/2 and 0=1r/2, the aforedescribed circuit will generate accurately the value of sin 19 for any 6 lying within these limits. As thus far described, the construction and operation of the circuit is substantially identical with that of FIG. 6. In the circuit of FIG. 6, and its counterpart in FIG. 10, if the value of the independent variable input 0 is allowed to exceed the limits 1r/ 2 and 11'/ 2, then output of the device, i.e., terminal 34 in FIG. 6 or terminal 80 of its counterpart in FIG. 10, would go very highly negative or positive until the amplifier saturates and thus gives an erroneous reading. The reason for this erroneous reading is that the maximum voltage which the device, as thus far described, can supply at terminal 74 in FIG. 10, for example, is 1r/2 or +1r/ 2 and this is sufficient to balance at junction 84 only 1r/2 or +1r/ 2 originating at terminal 82. If the difference between these two voltages appearing at 84 is not very close to zero, the tremendous amplification of amplifier 78 causes its output at 80 to rise until the amplifier saturates.

To extend the limits of the function would require some modification which would cause the output at terminal 80, which is, for example say +1 when 0 is to decrease when 0 increases to, say 93, until it reaches the same value that it had when 0 was 87, since sin (90 +3 :sin (90--3). In other words, the circuit of FIG. 6 and its counterpart in FIG. 10 can be made to produce a correct value for the sine of 6 with 0 equal to, say 93 if the effective 0 input were made 87 or in general if the effective input were reduced to a value 02(01r/ 2). This is accomplished in FIG. 10 by adding the two additional branches 86 and 88 to be used under appropriate circumstances to contribute to the voltage at terminal 76.

. The operation of the circuit can easily be understood by reference to FIG. 11 in which the solid line representation is a graph of effective input to junction 84 in FIG. 10 versus 0, which latter is applied to terminal 82. As the input of at terminal 82 goes from -1r/2 to 1r/2 the effective input at junction 84 must go from 1r/2 to 1r/ 2 and it does so, as illustrated in FIG. 11 by the line segment PQ, by virtue of the operation of the circuit heretofore described as the counterpart of FIG. 6. As 0 increases beyond 1r/ 2 and the input 6 at terminal 82, designated as -0 becomes more negative than 1r/ 2, it is required for the effective input at 84, designated as 0 to decrease in absolute magnitude to the value given by the equation 9 =0 +2(0 1r/ 2). The reason for this can be seen from an example using actual numbers. When say 0 =87, the output at terminal 80 is sin 87. When 0 =9O, the output at terminal 80 is sin 90. However, if 0 should be allowed to become more negative to say 93 then the system, which is built to work only within the limits 1r/2 to 'n'/ 2, cannot handle the -93 voltage and, so to speak, goes berserk yielding an output at 80 representing saturation of amplifier 78. But, observing that sin 93=sin 87, ,it is apparent that if, when 0 ='93, 0 can be made equal to 87, then the apparatus, which is fully capable of handling a voltage of 87 at terminal 84 without going berserk, will yield at terminal 80 a voltage equal to sin 87. This latter, of course, is numerically equal to sin 93 so that the apparatus is, in effect, handling a voltage input at 82 representing 0 1r/2.

It should be noted that the general requirement, previously stated, that for 0 1r/ 2, 6

must: -032+2(6 2-7r/2) is represented in the preceding numerical example thus:

To accomplish this requirement means contributing, at the time when t9 1r/ 2, a component at 84 which will add, to the component at 84 due to 0 the effect of 2( 0 1r/ 2) applied through an input resistor equal in size to 85. This added component arrives from the network comprised of branches 86 and 88. The same voltage 6 applied to terminal 82 is always simultaneously applied to terminal 90. When 0 at terminal 90 is more negative than 7r/2, the potential of the cathode of diode 94 is lowered below that of its place and hence diode 94 conducts, causing a current to flow in branch 86 whose magnitude is proportional to -0(1r/2) divided by 11 /2. This, in effect, contributes at junction 76 a potential of 2(0+1r/ 2) which, in passing through amplifier 66, changes its sign and, since resistor 95 equals resistor 85, appears at terminal 84 as, effectively, 2(01r/2), compared to the 0 at the same terminal contributed from terminal 82. The net or effective input, then, at terminal 84 upon initiation of the operation is 0+2(0-1r/2)=01r. If, as in the aforementioned example, 0:93 then the net effective input at terminal 84 would correspond to 93-1r=87, a magnitude which is within the limits of 1r/2 to -+1r/ 2 under which the circuit is capable of giving correct results. The production of the proper effective input at terminal 84 for the region 1r/2 e 1r/2 isshown graphically in FIG. 11 by the dotted line segment PR, representing the contribution from 0 the dash-dot line segment ST, representing the contribution from branch 86 equal to 2(9 1r/2); and the solid line segment PU representing the sum of the two contributions at terminal 84.

An analogous situation occurs with conduction in branch 88 when -31r/2 6 -1r/2. This is shown graphically in FIG. 11 by line segments QV and LM which add to produce QN.

This circuit can be used, by throwing switch 97 to the 1r/ 2 position, for generating cos 6 for the limits n- 021r. But, of course, it is operable only within these limits for the cos 6 (and,-31r/ 2 953117 2 for the sine) for the reason that these limits are necessary, with this circuit, to maintain the effective net input at 84 between 1r/2 and 1r/2. If 6 should exceed 31r/ 2, e.g., should be 271, then the net effective input at 84 would be 0-1r=271180=91 which is beyond the operating limits of the circuit. However, further extension beyond the range 31r/2 to 31r/ 2 for the sine and 1r to Zwr for the cosine is, of course, possible using the illustrated principle, i.e., by energizing appropriate circuits whenever the absolute magnitude of 0 exceeds 31r/2, 51r/ 2, etc., so that the effective input at 84 is always maintained in the range 1r/2 to +1r/ 2.

The circuit of FIG. 7 using Method B can also be modified to extend the range of 0. FIG. 12 illustrates such a modification showing one particular embodiment. When operating in the range of 1r/ 2 0 1172, the circuit compares 0 applied at terminal 102 with the time representation z(t) of the arc sine function applied at terminal 104 and, on the basis of the comparison, selects, in a manner similar to the operation of the circuit of FIG. 10, either +1 or --1 from terminals 106 or 108 as the value of the auxiliary variable. The auxiliary variable is averaged by the filter 110 yielding sin 0 at output terminal 112. If 0 exceeds 1r/2, diode 114 conducts and produces as the effective input at terminal 116 the sum of 9+2(0-1r/ 2) :6-11, the :first term on the left hand side being due to branch 118 and the second term being due to branch 120. This is so because resistor 119 is twice as large as resistor 121. So long as 0 31r/ 2, the quantity 0-1r effectively applied at 116 remains within the 1r/2 to 1r/2 range of effective inputs within which the circuit gives correct results. Similarly, when branch 122 conducts and the circuit yields correct results for 0 -31r/2. If switch 124 is swung to the 1r/2 terrninal the circuit operates to generate cos 0 for As indicated in the discussion of FIG. 10, the circuit of FIG. 12 can, of course, be extended using the illustrated principle beyond the range 31r/ 2031rl 2 for the sine and 1r6 27r for the cosine.

Generation of sin 0, cos 0 with unlimited angular range.lt is often important in problems using angles to have an unlimited angular range when generating sine or cosine functions. The circuits of FIGS. 6, 7, 10, and 12 can be adapted to this requirement through the use of an auxiliary circuit. This auxiliary circuit makes use of dB/dt to produce an oscillation that sweeps through the restricted angular ranges of the sine and cosine generators (e.g., for one sine generator the range would be from -1r/2 to +1r/ 2) at a rate proportional to dO/dt. When d6/dt is constant this oscillation becomes an isosceles triangular wave. The circuit, when used for example to supply a sine generator, will then supply the sine generator with an input 0 which always lies between 1r/2 and '+7r/2 and at each instant has a value such that its sine is equal to the sine of the actual angle 0 (which is the actual machine variable) at that instant. That is, the circuit in a sense performs a function which results in the mathematical equivalent of converting the actual 0, no matter howlarge it may be, into an angle in either the first or fourth quadrants having an equivalent sine. The circuit performs this function without receiving (after initiation of its operation) any actual 6 input but by receiving merely actual dB/dt input which latter it integrates with respect to time in order to be able to sense increments of actual 0. A preferred embodiment of the auxiliary circuit is shown in FIG. 13.

The circuit of FIG. 13 comprises an operational amplifier 126 whose output at terminal 128 will ultimately be the desired 0 whose negative would be fed into, for example, terminal 29 of the sine generator of FIG. 6 or the like. The amplifier 126 is shunted by a condenser 130.

The capacitor-shunted amplifier 126, 130 is located in one branch 132 of a parallel circuit including another branch 134, which parallel circuit is connected in series with a pair of operational amplifiers 136 and 138. Amplifier 136 is shunted by alternatively operating branches 140 and 142, the former branch including a diode 144 and a voltage source such as a battery 146 of value 1r/2, and the latter branch including a diode 148 and a voltage source such as a battery 150 of such a value as to produce at terminal 152 a voltage of 1r/2 when branch 142 is conducting.

Branch 132 includes two resistors 154 and 156 of equal value at whose junction 158 is connected the output of a circuit yielding angular rate of change. This angular rate circuit receives at its input terminal 160 the quantity ds/dr, the time rate of change of the actual machine variable 6, which it can apply to junction 158 when diode 162 is conducting. Alternatively, when diode 164 is conducting, the angular rate circuit can apply d/dt to junction 158, the negative being obtained by simply passing dO/dt through the amplifier 166.

To initiate the operation of the circuit of FIG. 13, both 0 and dH/dt must be initially available but, after initiation of the operation, all that is needed is dB/dt and no further need exists for information as to the value of the actual machine variable 6' to enable the device to continue functioning. The operation of the device proceeds as follows. At time t=0, 0, the quantity appearing at terminal 128, is assumed to be 0(0). This value is established by applying, either automatically or manually, a voltage across capacitor 130, this being the initial value of the actual machine variable 9. This voltage can be applied by simply placing a battery of the correct value across the terminals of condenser 130, it being remembered that the potential at the summing junction 168 of the operational amplifier 126 is always substantially zero or ground. At the same instant that the initial value of 0 is applied across condenser 130, dB/a't is connected to terminal 160. At time t=0+5, the battery imposing 0(0) across condenser 130 is removed. While the battery was in position across condenser 130, the potential across the condenser was necessarily maintained constant. Upon removal of the battery, however, the amplifier 126 with its associated condenser acts as an integrator and begins to integrate its input voltage which is applied to one or both of its input resistors 154, 156. Assuming that O6(O) 7r/2, it will be intended for the integrator to add to the initial value 0(0) appearing at 128 the increment represented by the integral of dfi/dt over a period of time until the value of 0 at 128 reaches 1r/ 2. To insure that the initial operation is started in the right direction to perform this addition, it is required that, at the start of the operation, a positive input should exist at input terminal 188 to the amplifier 136. This can easily be accomplished by throwing the switch 188 to a source such as 186 of positive potential, which could be for example merely one volt, at the instant of the start of the operation and then throwing it back into the solid line position very rapidly, using a make-before-break switch if desired. The reason for applying an initial positive potential at 188 can be seen from the following analysis.

With, say, +80 at 128 from the starting battery applied across 130, there would be experienced at summing junction 174 the effect of +80 from 128 plus the effect transmitted from terminal 152. At 152 there will, however, be a voltage of -1r/2 produced by virtue of the following sequence of events. When terminal 188 is connected to the positive battery source 186, the output of amplifier 136 at 152 will be negative. By virtue of battery 150 and diode 148, it is held at a negative level of -1r/ 2. Therefore, at summing junction 174 there will be felt the efiect of say, +80 from 128 combined with 90 from 152 giving a net negative elfect at 174 which will emanate with a change of sign as a positive voltage at 172. This posi- 20 tive voltage at 172 is fed, through resistor 189, into summing junction 170, thus maintaining the circuit in a stable state since the positive starting voltage at 188 from the battery 186 was precisely the sign required to produce a positive voltage at 172 to be fed into 188 so that the device will be self-maintaining.

With -1r/2 appearing at junction 152, as just described, the potential at junction 158 will be 1r/4 since resistors 154 and 156 are equal and the potential at 168, as previously indicated, is substantially zero. di9/dt is assumed to have a value between zero and 1r/4. The presence of 1r/4 at junction 158 therefore causes diode 164 to conduct, thereupon clamping the voltage at junction 158 at the level of d6/dt which might be at, say 40 volts. With 40 volts at terminal 158, the integrating amplifier 126 will integrate this voltage continuously as long as it is applied at terminal 158, resulting in an increase in the positive voltage at terminal 178 and hence, at 128. When the voltage at 128 has reached 'rr/Z, a change will occur. As soon as the voltage at 128 exceeds ever so slightly 'rr/Z, the net effect at junction 174 will fiip from negative to positive. For example, +91 at junction 128 combined with 1r/2 from junction 152 will yield a net effect at 174 of +1. This positive voltage at 174 changes its sign by passing through amplifier 138, and the voltage at 172 will then be negative. A negative voltage at 172 fed into junction 170 will produce a positive voltage at 152, which positive voltage will be fixed at 1r/2 by the limiting eliect of branch having battery 146 and diode 144.

As soon as +7r/ 2 appears at 152 this will tend to produce at junction 158 a potential of +7r/4 which instantly stops diode 164 from conducting and causes diode 162 to conduct, transmitting to junction 158 the voltage dO/dt originating at terminal 160. Assuming, as previously stated, that d0/dt=+40, merely for example, this positive voltage at 158 will appear as a negative voltage at 132, having changed its sign by passing through amplifier 126, and this negative voltage will be integrated by the integrator amplifier 126, thus reducing the positive potential at terminal 178 from 1r/2 down through zero until it reaches 'n'/2. When the 0 output at 128 attempts to go more neagtive than this -7r/2 value, another change occurs the reverse of that previously explained. That is, the say 91 at 128 overbalances the ]11'/2 at 152 and thus produces a not negative voltage at 174 which emanates as a positive voltage at 172 and enters junction as a positive voltage causing a negative voltage to appear at 152, which negative voltage is fixed at 1r/2 by limiter 142. The 71'/ 2 voltage at 152 tends to produce a potential at 158 of 1r/4 which stops the conduction of diode 162 and starts again the conduction of diode 164, thus clamping terminal 158 at the potential d0/dt. The circuit is now back to the same type of condition that obtained at the original start where the output at 132 was positive. This positive voltage is integrated and thus raises the potential at 178 and 128 from 1r/2 through zero to +1r/ 2, at which time the sequence of events repeats itself and then goes on indefinitely repeating.

It should be noted that the integrator circuit is a linear integrator which causes the voltage across condenser 130, which is to say the output 0 at 128, to rise and fall linearly with time between the limits of -1r/2 and +1r/ 2. Assuming that dB/dt is a constant, the graph of 0 output at 128 versus time will be the triangular wave shown in FIG. 13. From the foregoing, it is apparent that the circuit of FIG. 13 produces an output 0 at terminal 128 which lies always between 1r/2 and +1r/2 and Will be of such a value that its sine will always equal the sine of the actual machine variable 0 as the machine variable 0 increases indefinitely from any initial value lying between 1r/2 and +1r/ 2. The circuit accomplishes this by being supplied only initially at terminal 178 with only the initial value of the actual machine variable 0 and by being 

3. AN APPARATUS FOR GENERATING THE PRODUCT OF A FIRST INDEPENDENT VARIABLE BY THE SINE OF A SECOND INDEPENDENT VARIABLE COMPRISING A COMPARATOR; MEANS FOR SUPPLYING TO SAID COMPARATOR AN INPUT WHOSE ABSOLUTE VALUE REPRESENTS THE ABSOLUTE VALUE OF SAID SECOND INDEPENDENT VARIABLE; MEANS FOR SUPPLYING TO SAID COMPARATOR A PERIODIC TIME REPRESENTATION OF THE ARC SINE FUNCTION FOR COMPARISON WITH THE VALUE OF SAID SECOND INDEPENDENT VARIABLE AN AUXILIARY FUNCTION GENERATOR CONTROLLED BY SAID COMPARATOR FOR GENERATING AN AUXILIRY FUNCTION HAVING ABLY TWO VALUES, ONE OF SAID VALUES BEING THE VALUE OF SAID FIRST INDEPENDENT VARIABLE AND THE SECOND OF SAID VALUES BEING THE NEGATIVE OF SAID FIRST INDEPEDENT VARIABLE; AND MEANS FOR AVERAGING THE OUTPUT OF SAID AUXILIARY FUNCTION GENERATOR. 